Continuity and canceling operators of order n on $${\mathbb {R}}^n$$

Abstract

AbstractWe prove that for elliptic and canceling linear partial differential operators $${\mathbb {B}}$$B of order n on $${\mathbb {R}}^n$$Rn, continuity of a map u can be inferred from the fact that $${\mathbb {B}}u$$Bu is a measure. We also prove strict continuity of the embedding of the space $${\text {BV}}^{\mathbb {B}}({\mathbb {R}}^n)$$BVB(Rn) into the space of continuous functions vanishing at infinity. Here, $${\text {BV}}^{\mathbb {B}}(\varOmega )$$BVB(Ω) denotes the space of vector fields $$u\in {\text {W}}^{n-1,1}(\varOmega )$$u∈Wn-1,1(Ω) such that $${\mathbb {B}}u$$Bu is a finite vectorial measure on the open set $$\varOmega \subset {\mathbb {R}}^n$$Ω⊂Rn. For cubes $$Q\subset {\mathbb {R}}^n$$Q⊂Rn, the class of operators $${\mathbb {B}}$$B such that $${\text {BV}}^{\mathbb {B}}(Q)\subset {\text {C}}({\bar{Q}})$$BVB(Q)⊂C(Q¯) is also characterized.